Dr. Marijn Heule

University of Texas, Austin

Everything's Bigger in Texas: The Largest Math Proof Ever

Zusammenfassung

The Boolean Pythagorean Triples problem has been a long-standing open
problem in Ramsey Theory: Can the set N = {1, 2, 3, ...} of natural numbers be
partitioned into two parts, such that neither part contains a triple (a, b, c) with a^2 +
b^2 = c^2 ? A prize for the solution was offered by Ron Graham decades ago. We
show that such a partition is possible for the set of integers in [1,7824], but that it is
not possible for the set of integers in [1,M] for any M > 7824. Of course, it is
completely infeasible to attempt proving this directly by examining all 2^M possible
partitions of [1,M] when M = 7825, for example. We solve this problem by using the
cube-and-conquer paradigm, a parallel satisfiability solving method that achieves
linear time speedups on many hard problems. An important role is played by
dedicated look-ahead heuristics, which allowed us to solve the problem on a cluster
with 800 cores in about two days. Due to the general interest in this mathematical
problem, our result requires a formal proof. Exploiting recent progress in
unsatisfiability proof checking, we produced and verified a clausal proof of the
smallest counterexample, which is almost 200 terabytes in size. From this we
extracted and made available a compressed certificate of 68 gigabytes, that allows
anyone to reconstruct the proof for checking. These techniques show great promise
for attacking a variety of challenging problems arising in combinatorics and computer
science.

Vortragender

Dr. Marijn J.H. Heule is a Research Scientist at the University of
Texas at Austin. He received his PhD at Delft University of Technology in the
Netherlands in 2008. His research focuses on solving hard combinatorial problems in
areas such as formal verification, number theory, and combinatorics. Most of his
contributions are related to theory and practice of satisfiability (SAT) solving. He has developed award-winning SAT solvers (see http://satcompetition.org/), and his
preprocessing techniques are used in most state-of-the-art SAT solvers.<br><br>
More recently, Dr. Heule has concentrated on two major challenges for SAT solving:
1) exploiting the potential of high-performance computing; and 2) validating the
results of SAT solvers and related tools. Dr. Heule developed a new parallel SAT
solving paradigm, called cube-and-conquer, which enables linear time speedups on
many hard problems. His first publication on cube-and-conquer won the best paper
award at HVC 2011. Treengeling, a solver by Armin Biere based on that paradigm,
solved most benchmarks during SAT Competition 2013.<br><br>
The increasing complexity of SAT solvers and related tools makes it more likely that
these tools contain bugs. Dr. Heule has designed a new proof format and
implemented a fast, corresponding proof checker for SAT and QBF solvers. Prooflogging
in his format has been mandatory for the SAT Competitions since 2013,
thereby increasing the confidence that tools produce correct results.<br><br>
Dr. Heule has published over 40 technical peer-reviewed papers. Dr. Heule is one of
the editors of the Handbook of Satisfiability. This 900+ page handbook has become a
standard for the SAT community, and it is a tremendous resource for future scientists.
Dr. Heule is an Associate Editor of the Journal on Satisfiability, Boolean Modeling
and Computation and was a co-chair of the International Conference on Theory and
Applications of Satisfiability Testing in 2015 in Austin.